Monday, February 5, 2018

Counting down from infinity

In one version of the Kalaam argument, Bill Craig argues against forming an infinite past by successive addition by asking something like this: Why would someone who had been counting down from infinity have been finished today rather than, say, yesterday? This argument puzzles me. After all, there is a perfectly good reason why she finished today: because today she reached zero and yesterday she was still on the number one. And yesterday she was on one because the day before she was on two. And so on.

Of course, one can object that such a regress generates no explanation. But then the Kalaam argument needs a Principle of Sufficient Reason that says that there must be explanations of such regressive facts and an account of explanation according to which the explanations cannot be found in the regresses themselves. And with these two assumptions in place, one doesn’t need the Kalaam argument to rule out an infinite past: one can just run a “Leibnizian style” cosmological argument directly.

9 comments:

As Edward Feser argued against Arif Ahmed, you may even need a principle of sufficient reason to conduct reasoning itself. I don't see how Bill's explanation would require the PSR any more than anything else that needs to be explained. If you don't accept the PSR, you might still reason that a vicious infinite regress is unacceptable, which is what you get when you say "the countdown is at 0 today because it was at 1 yesterday, because it was at 2 the day before..."

I think Craig's point is that the answer we give for "why haven't they finished X yet" is always something like "they haven't been doing it long enough". Such an answer is unavailable in the case of infinite time. 100 years ago, they had already had an actually infinite amount of time to finish. That is to say, if they count one number every second, they have already had an actually infinite number of seconds (coextensive with the negative numbers), so they should have finished 100 years ago. Same goes for 1,000 or a billion or even an infinite number of years ago. So, it is inexplicable that they just now finished.

Besides, if you are saying that each negative number he has counted corresponds to an appropriate past moment (say, a previous second), and that that is true all the way back, forever, then one can just rephrase it in terms of many counting men. Why is it that different counting men end at different times (100s or billions of years apart), and yet they counted exactly the same set of numbers one second at a time? After all, there is a possible counting man who only finishes a billion years from now. And there will be a negative number for each second of his past counting, just as there is for every second of the past counting of the counting man who finishes today. People who are counting the same set of numbers, exactly as quickly as each other, shouldn't finish billions of years apart.

Of course, as Bill Craig insists, the Kalaam argument only works for those who accept an A-theory of time. Perhaps a B-theorist, like yourself, can make sense of a number corresponding to every past event, and yet there being no beginning. But, Craig is specifically using "counting" to denote the succession of comings-into-being and goings-out-of-being.

I think on this formulation, the story becomes just a slight twist on the intuitions already there in Hilbert's Hotel, where one can ask exactly analogous questions. Take a bunch of infinite hotels, with at most one person to the room, and ask: Why do some of the hotels have all the rooms filled and others not, given that they all have the same number of rooms?

"People who are counting the same set of numbers, exactly as quickly as each other, shouldn't finish billions of years apart."

That's false even in the finite case. It only becomes true if one adds the qualifier: "if they started at the same time." But in the infinite case they didn't start at the same time, because they didn't start.

The principle you offer would need to be rephrased, then. I think it would be an awkward rephrasing, and probably not very different from the sorts of principles that directly yield Hilbert's hotel, like: people who put the same number of items per room in different rooms shouldn't fill sets of rooms one of which is a proper subset of the other. In other words, you get a principle that is plausible in the finite case but pretty directly begs the question against Dedekind infinite sets (i.e., ones that are the same size as a proper subset).

I think the point of Hilbert's Hotel is to demonstrate that, though this is perfectly consistent as pure mathematics, it would be incoherent and absurd if actualized in reality.

That being said, the counting man doesn't reduce to Hilbert's Hotel. It's making the further point that potential infinities (what is symbolized by a lemniscate) can never be fully completed by successive additions (which transfinite arithmetic and set theory say absolutely nothing about, and which had better be true if basic calculus is going to work). If one insists that an actually infinite sequence has actually elapsed prior to now, the question "why didn't it finish 100 years ago?" seems not to have an answer. The only answer such a question normally has is "they hadn't had sufficient time yet". But such an aswer doesn't work with an infinite past series. They've had enough time to get to zero since infinitely long ago.

I feel like I'm missing something here. Craig is using this illustration to point out that, before the person counting from infinity reaches ANY discreet number, they would have had to completely count through an infinite set to reach that number. But if they have already counted through an infinite set, then they should have reached that number already. So the idea of counting down from infinity implies both that no discreet number could ever be reached and that all discreet numbers have already been reached. It's almost a blatant self-contradiction, even if you look past the absurdity of counting through an infinite set through successive addition. What am I missing here?

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